\subsection{}
\index{Feshbach Resonance!Narrow}
One key ideas to hide all detail in high-energy (not clear yet) into some constant by renormalization.  All coupling are short-range (within potential range $r_{c}$), and we should hide it.  On the other hand, the detuning $\delta=E^{0}-\eta$ is not necessarily in this high-energy region for narrow resonance as it might smaller than Fermi energy.  In \eef{eq:20100915:tu}, detuning $\delta=E^{0}-\eta$ can get so large in part of fermi sea that $\tilde{U}$ becomes small.  We need to figure out a way to renormalize out the k-denpendence in $U_{\vk\vk'}$ and $Y_{\vk\vk'}$ but keep detuning around.  

\eef{eq:20100915:gap} is very similar to two-body \sch equation in zero energy.  However, the narrow resonance is probably more relevant with two-body \sch equation in finite energy.  How to bring it in?

\subsection{Chemical potential}
\emph{Chemical potential is determined in different ways between narrow or broad resonance.  }In broad case, it is determined mostly by open-channel \eef{eq:20100915:gapa}; in narrow case, chemical potential is determined  except the very BCS end,  by where the close-channel bound-state level sits relative to Fermi sea.  In the extreme narrow case (without open-channel interaction as \cite{GurarieNarrow}), the level is exactly where chemical potential sits, cutting Fermi sea, depleting everything above in open-channel and putting them into close-channel.   

By  putting the chemical potential $\mu$ in the proper position, we can see the narrow/broad resonance even in the four species case.  In Eq. (\ref{eq:20100915:t0}-\ref{eq:20100915:tk}), there are two many-body effects: $\sqrt{1-4G_{\vk}^{2}}$ from three-species Pauli exclusion; chemical potential in detuning term $E^{0}-\eta+\mu$, which is common in either three or four-species case.  And the later reduces to $\mu=0$ \footnote{In BEC side, $\mu<0$ and is controlled by mostly two-body attraction.  But that is not proper for real two-body limit, which should be $\mu=0$.}in zero-density which is two-body case. 
\begin{figure}[hhtb]
	\centering
		\includegraphics[width=.50\textwidth]{image/FeshbachPotential}
	\caption{Feshbach Resonance Potential\label{fig:FeshbachPotential}}	
\end{figure}

Imaging we start fairly far away from resonance (BCS side, $\delta=\eta-E^{0}>0$), and increase the density, in the beginning, $\mu$ is negligible, and inter-channel coupling term $\frac{Y_{\vk\vk''}Y_{\vk''\vk'}}{2(E^{0}-\eta+2\mu)} $ is small; as $\mu$  increases, $-(\delta-\mu)$ gets closer and closer to zero, and this terms increases until the part of the Fermi sea gets into resonance.  Then most of adding fermions go into close-channel.  





\subsection{Renormalization of gap equation}
There are more than one options to renormalize gap equation \eef{eq:20100915:onechannel}.  The part that needs to be renormalized out is  high energy summation of $\sqrt{\frac{(1-4G_{\vk'}^{2})}{{(\xi^{ab}_{\vk}+  G_{\vk}^2\eta)^{2}+\Delta_{\vk}^{2}}}}$, it approaches $\nth{\epsilon_{\vk}}$ in high energy.  Several sightly different physical quantities have the summation of the same high energy limit.  
\begin{enumerate}
\item The process used in Eq. (\ref{eq:20100915:t0}-\ref{eq:20100915:tk}).  This leads to the zero energy T-matrix of reduced DoS with factor $\sqrt{(1-4G_{\vk'}^{2})}$, with chemical potential in the detuning.  
\begin{equation}\tag{\ref{eq:20100915:renormGap}}
\nth{\tilde{t_{0}}(\mu)}=\sum_{\vk}\sqrt{(1-4G_{\vk}^{2})}
\br{\nth{\epsilon_{\vk}}-\nth{\sqrt{{(\xi^{ab}_{\vk}+  G_{\vk}^2\eta)^{2}+\Delta_{\vk}^{2}}}}}
\end{equation}
\begin{gather}
\tilde{t_{0}}(\mu)=\br{1-\tilde{U}\tilde{ K}}^{-1}\tilde{U}\tag{\ref{eq:20100915:t0}}\\
\tilde{U}_{\vk\vk'}=\nth{2} \br{U_{\vk\vk'}+\frac{Y_{\vk\vk''}Y_{\vk''\vk'}}{2(E^{0}-\eta+2\mu)}}\tag{\ref{eq:20100915:tu}}\\
\tilde{K}=\frac{\sqrt{1-4G_{\vk}^{2}}}{\epsilon_{\vk}}\delta_{\vk\vk'}\tag{\ref{eq:20100915:tk}}
\end{gather}
\item We can also notice that $G_{\vk}\rightarrow0$ at high energy.  So we can simply takes the normal zero-energy T-matrix with detuning related to chemical potential.  
\begin{equation}\label{eq:20101004:renormGap1}
\nth{{t_{0}}(\mu)}=\sum_{\vk}
\br{\nth{\epsilon_{\vk}}-\frac{\sqrt{(1-4G_{\vk}^{2})}}{\sqrt{{(\xi^{ab}_{\vk}+  G_{\vk}^2\eta)^{2}+\Delta_{\vk}^{2}}}}}
\end{equation} 
\begin{gather}
{t_{0}}(\mu)=\br{1-\tilde{U}\tilde{ K}}^{-1}\tilde{U}\label{eq:20101004:t01}\\
\tilde{U}_{\vk\vk'}=\nth{2} \br{U_{\vk\vk'}+\frac{Y_{\vk\vk''}Y_{\vk''\vk'}}{2(E^{0}-\eta+2\mu)}}\label{eq:20101004:tu1}\\
{K}=\nth{\epsilon_{\vk}}\delta_{\vk\vk'}\label{eq:20101004:tk1}
\end{gather}
Here Eqs. (\ref{eq:20101004:t01}-\ref{eq:20101004:tk1}) follows the same two-body formula for zero-energy T-matrix element.  However, the detuning is shifted by a many-body quantity $\mu$ that should be determined by solving gap equation with numberequation.    

\item  Alternatively, we notice that $\xi_{\vk}=\epsilon_{\vk}-\mu$, high energy limit can also be written as 
$\nth{\epsilon_{\vk}-\mu}$.  This leads to the T-matrix at energy $\mu$, the same detuning as before.  
\begin{equation}\label{eq:20101004:renormGap2}
\nth{{t_{\mu}}(\mu)}=\sum_{\vk'}
\br{\nth{\epsilon_{\vk'}-\mu}-\frac{\sqrt{(1-4G_{\vk'}^{2})}}{\sqrt{{(\xi^{ab}_{\vk}+  G_{\vk}^2\eta)^{2}+\Delta_{\vk}^{2}}}}}
\end{equation} 
\begin{gather}
{t_{\mu}}(\mu)=\br{1-\tilde{U}\tilde{ K}}^{-1}\tilde{U}\label{eq:20101004:t02}\\
\tilde{U}_{\vk\vk'}=\nth{2} \br{U_{\vk\vk'}+\frac{Y_{\vk\vk''}Y_{\vk''\vk'}}{2(E^{0}-\eta+2\mu)}}\label{eq:20101004:tu2}\\
{K}=\nth{\epsilon_{\vk}-\mu}\delta_{\vk\vk'}\label{eq:20101004:tk2}
\end{gather}
The advantage of this is that introduce the effective range $r_{0}$ for finite energy T-matrix. 
\end{enumerate}
\emph{Note that Eqs. (\ref{eq:20101004:renormGap1}, \ref{eq:20101004:renormGap2}) cannot take the approximation of $G_{k}\approx{G_{0}}$ because that leads the summation diverged.  One has to take a more realistic $G_{k}$ that approach 0 at high momentum.  }
\subsection{Pauli exclusion factor  $\sqrt{(1-4G_{\vk'}^{2})}$ }
It seems that numerous Gap equations above has the Pauli-exclusion factor $\sqrt{(1-4G_{\vk'}^{2})}$.  Generally, 
\begin{equation}
G_{k=0}^{2}\sim{\frac{N_{close}}{N_{0}}\frac{r_{close}^{3}}{a_{0}^{3}}}
\end{equation}
where $r_{close}$ is close-channel molecule size, $N_{0}$ is the total number of fermions, $N_{close}$ is the number in close-channel, $a_{0}$ is the average particle distance.  This quantity is related the detail of close-channel bound-state.  We should relate it to some experimental available quantities.  In the above expression, factor $r_{close}^{3}/a_{0}^{3}$ relates to the two-body physics, while $N_{close}/N_{0}$ relates to many-body physics which probably needs to be solved consistently with the many-body equation .  

\subsection{More on chemical potential shift for detuning}
Let us image a sweep of $\delta$ from BCS end.  At the very BCS end, $\delta$ is positive and large than $\mu\approx{}E_{F}$ (Fig. \ref{fig:narrowFR:aboveSea}). Resonant term in interaction is relatively small and close-channel weight is small too.  We are very well in the  weak-attraction (slightly enhanced by resonance) BCS-like state in open-channel.  However, the detuning is shifted by $\mu\approx{}E_{F}$, so the resonance is reached earlier, and the larger the density, the earlier it does.  

As the detuning gets close to Fermi surface, chemical potential decreases from $E_{F}$. For narrow resonance, $E_{F}$ is large than the resonance energy scale.  The resonance is reached probably before detuning reaches Fermi surface.  If we ignore the shift due to open-channel intra-channel coupling for the moment, the resonance is very close at the point where $\delta=E_{F}$.  After $\delta$ drops into Fermi sea (Fig. \ref{fig:narrowFR:inSea}), chemical potential $\mu$ tracks the close-channel bound-state closely and the denominator of resonant term $-(\delta-\mu)$ keeps very small, and in some sense, the Feshbarch resonance is enhanced by the many-body  physics.  Open-channel still has energy advantages below $\mu$, so both channels are important.  

After $\delta$ drops below zero (Fig. \ref{fig:narrowFR:belowSea}), chemical potential still tracks $\delta$, but now most weight is in close-channel and the bound-state is mostly made of close-channel.  

\subsection{Number equation}
For number equation \eef{eq:20100909:number}, more approximation can be made in Feshbach resonance.  Close-channel component $G_{k}$ is much extended in k-space due to the tight-binding, and $F_{k}$ is significant up to the order of Fermi energy $E_{F}$.  It seems OK to assume that most weight of close-channel lies beyond $E_{F}$, so when $\epsilon_{k}\gg{E_{F}}$, the integrand becomes simply $G_{k}^{2}$, the summation gives $N_{close}$.  Therefore we have 
\begin{equation}\label{eq:20101004:number}
N_{open}=\sum_{\vk}{}^{'}\left(1-\sgn_{k}\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}\right)-G_{\vk}^{2}
\end{equation} 
Here the summation only goes up to the order of $E_{F}$

\subsection{summary}
Now all summation goes in order of Fermi energy $E_F$, so we can replace close-channel component $G_k$ with $G_0=G_{k=0}$
\begin{equation}
F_{\vk}=\frac{\Delta}2\sqrt{\frac{(1-4G_{\vk}^{2})}{(\epsilon^{ab}_{\vk}-2\mu+  G_{\vk}^2\eta)^{2}+\Delta^{2}}}
\end{equation}
Gap equation 
\begin{equation}\label{eq:20101004:gapS}
\nth{{t_{0}}(\mu)}=\sum_{\vk}
\br{\nth{\epsilon_{\vk}}-\frac{\sqrt{(1-4G_{\vk}^{2})}}{\sqrt{{(\epsilon^{ab}_{\vk}-2\mu+  G_{\vk}^2\eta)^{2}+\Delta^{2}}}}}
\end{equation} 
\begin{gather}
{t_{0}}(\mu)=\br{1-\tilde{U}\tilde{ K}}^{-1}\tilde{U}\label{eq:20101004:t011}\\
\tilde{U}_{\vk\vk'}=\nth{2} \br{U_{\vk\vk'}+\frac{Y_{\vk\vk''}Y_{\vk''\vk'}}{2(E^{0}-\eta+2\mu)}}\label{eq:20101004:tu11}\\
{K}=\nth{\epsilon_{\vk}}\delta_{\vk\vk'}\label{eq:20101004:tk11}
\end{gather}
Here Eqs. (\ref{eq:20101004:t011}-\ref{eq:20101004:tk11}) follows the same two-body formula for zero-energy T-matrix element.  However, the detuning is shifted by a many-body quantity $\mu$ that should be determined by solving gap equation with number equation.  
And number equation
\begin{equation}\tag{\ref{eq:20100909:number}}
N=\sum_{\vk} \left(1-\sgn_{k}\sqrt{1-4 F_{\vk}^2-4 G_{\vk}^2}\right)
\end{equation} 

\subsection{Tony's comment}
Upon hearing my argument about chemical potential and scenarios in Fig. \ref{fig:narrowFR}. He thought it is like impurities in semiconductor band, its level with the band and affect the chemical potential. He suggested me to check text from Kettel.  

He also mentioned about three-species and four-species.  In four-species two species flip and needs two powers and in three-species case only one flips, so one power.  Therefore four-species are probably narrower.  

\subsection{$G_0$ and $\eta$}
In the full region, $\eta$ should be close to the binding-energy of the close-channel bound-state all the time.  The sweep in detuning should always be smaller than $\eta$ in order.  So maybe it is fine to take it as constant of binding energy $E^0$.  However, this quantiy is a two-bdoy quantity and very specific for different case. 

$G_0=\alpha\phi_0$, where $\phi_0$ is normlized w.f. of two-body close-channel bound-state, strictly two-body.  Also, it is different from case to case. Its Pauli exclusion effect only shows up in many-body physics. It is determined by the real size of close-channel bound-state and therefore relates to binding energy $E^0$.  

For the same $a_s$ or $t_0$, $G_0$ or $\eta$ can take different value for different resonance.  Maybe we can take it as a different exogenous parameter. But we still need to find a way to estimate/deduce it from measurable quantities.  It is possible that they only relate to many-body quantities (same number for different densities maybe?)  because they are pertinent in many-body physics.